It would be very helpful if we could enable that search uses the word below the cursor and pre-populates the "find what" string.

The logic is simple:
- Find Next / Previous act as today
- If the cursor is above a word, then the search dialog pre-populates this word in the "find what" string
- If the cursor is in between white spaces, then the search dialog act as today

If you select the word it will use it as a find expression.
I was trying to think how often I search for the word I currently happen to have the caret on and
I personally think that I most often search for something unrelated to the current word.

I agree with Jehoschua - usually when im invoking find I am looking for that particular word. If its a function or variable that's easy enough since I can right click on it and look for all the declarations of that variable/function but if I need to find something that is NOT one of those items - it would be useful. Life doesn't end if this feature doesn't get added - its a nice to have though for sure.

Two swallows do not make a summer. Let's get some more opinions here. And please, no made up percentages. I very much doubt you did a comprehensive tally on how often you use the current word under the cursor or not while you were sleeping.
We read your post, so please let others post.

Can't say I care either way or can follow the issue:
- If I've already selected a section of code, "CTRL" + "C" is easy enough to run.
- If, when the search window pops up, the currently marked text is also already marked - as it is - I can type in a new text as if it were empty, when I try looking for something else.
- When I want to look for the next instance of the current selection, I can easily mark it and press "CTRL"+"F3"
The current state is perfect, as far as I am concerned.

If that wasn't actually the functionality you were complaining about, Thomas, then I don't really understand how you use it.
Maybe do a short screen recording to emphasize it, because I really think the two of us use PowerShell Studio very differently from each other.